A container has 0.6 moles of a monoatomic gas at 27°C. If 0.09 moles of the same gas at 227°C is added to it, the final equilibrium temperature of the container will be around.

Options

A. `50^(@)C`
B. `70^(@)C`
C. `30^(@)C`
D. `100^(@)C`

Answer

To find the final equilibrium temperature of the container after mixing two amounts of a monoatomic gas at different temperatures, we can use the concept of internal energy and the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Identify the given data
- Moles of first gas (n1) = 0.6 moles
- Temperature of first gas (T1) = 27°C = 27 + 273 = 300 K
- Moles of second gas (n2) = 0.09 moles
- Temperature of second gas (T2) = 227°C = 227 + 273 = 500 K

### Step 2: Calculate the total number of moles
The total number of moles after mixing:
\[ n_{total} = n_1 + n_2 = 0.6 + 0.09 = 0.69 \text{ moles} \]

### Step 3: Write the expression for internal energy
The internal energy (U) of a monoatomic gas can be expressed as:
\[ U = \frac{3}{2} nRT \]
Where:
- n = number of moles
- R = universal gas constant
- T = temperature in Kelvin

### Step 4: Calculate the internal energy of both gases
1. For the first gas:
\[ U_1 = \frac{3}{2} n_1 R T_1 = \frac{3}{2} \times 0.6 \times R \times 300 \]
\[ U_1 = 0.9 \times R \times 300 = 270R \]

2. For the second gas:
\[ U_2 = \frac{3}{2} n_2 R T_2 = \frac{3}{2} \times 0.09 \times R \times 500 \]
\[ U_2 = 0.135 \times R \times 500 = 67.5R \]

### Step 5: Write the equation for total internal energy
The total internal energy after mixing:
\[ U_{total} = U_1 + U_2 = 270R + 67.5R = 337.5R \]

### Step 6: Set up the equation for final temperature
The total internal energy can also be expressed in terms of the final temperature (Tf):
\[ U_{total} = \frac{3}{2} n_{total} R T_f \]
Substituting the values:
\[ 337.5R = \frac{3}{2} \times 0.69 \times R \times T_f \]

### Step 7: Simplify the equation
Cancel R from both sides:
\[ 337.5 = \frac{3}{2} \times 0.69 \times T_f \]
\[ 337.5 = 1.035T_f \]

### Step 8: Solve for Tf
\[ T_f = \frac{337.5}{1.035} \approx 326.0 \text{ K} \]

### Step 9: Convert to Celsius
To convert Kelvin to Celsius:
\[ T_f = 326.0 - 273 = 53.0 \text{ °C} \]

### Final Answer
The final equilibrium temperature of the container will be approximately **53°C**.

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